The aim in the current study is to examine the conceptualizations of zero in arithmetic operations among students with learning disabilities (LD) and no learning disabilities (N-LD). The similarities and differences in the understandings of students with LD and N-LD of zero in arithmetical operations will be discussed. The study is a multiple case…

Multiplicative reasoning is a key concept in elementary school mathematics. Item statistics reported by the National Assessment of Educational Progress (NAEP) assessment provide the best current indicator for how well elementary students across the U.S. understand this, and other concepts. However, beyond expert reviews and statistical analysis,…

During a time when students are accessing instruction virtually because of COVID-19, reflecting on the sixth of the Standards for Mathematical Practice (SMP 6) is a vital step for educators to take. Although the Common Core State Standards for Mathematics (NGA Center and CCSSO 2010) identifies SMP 6 as Attend to precision, associated expertise…

To meet increasingly complex mathematics standards in late elementary school, students must conceptually understand and be fluent in the operations of multiplication and division. This includes understanding the operations' inverse relation. The purpose of the study was to investigate the effects of alternating concrete-representational-abstract…

This study examined preservice elementary teachers' change in their understanding of fraction operations while taking a mathematics methods course focused on grades 3-5. The preservice teachers (n = 48) completed an assessment before and after the course's unit on the teaching and learning of fractions. Specifically, the preservice teachers were…

Theoretical analysis of whole number-based calculation strategies and digit-based algorithms for multi-digit multiplication and division reveals that strategy use includes two kinds of reasoning: reasoning about the relations between numbers and reasoning about the relations between operations. In contrast, algorithms aim to reduce the necessary…

Dyscalculia, or mathematics learning disorders, is currently known to be heterogeneous (Wilson & Dehaene, 2007). While various profiles of dyscalculia coexist, a general and persistent hallmark of this math learning disability is the difficulty in memorizing arithmetic facts (Geary, Hoard & Hamson, 1999; Jordan & Montani, 1997; Slade…

This article reports on a study in which researchers asked children to "make up as story and a picture about marbles for this number sentence: 3 x 5 = 15." Students in this study came from pre - dominantly low- to average-income families living in three distinct geographical areas within the United States. A similar division task was…

Algorithms and representations have been an important aspect of the work of mathematics, especially for understanding concepts and communicating ideas about concepts and mathematical relationships. They have played a key role in various mathematics standards documents, including the Common Core State Standards for Mathematics. However, there have…

Recent years have seen a quick expansion of tablet computers in households and schools. One of the educational affordances of tablet computers is using math apps to engage students in mathematics learning. However, given the short history of the mobile devices, little research exists on the effectiveness of math apps, particularly for struggling…

This study aimed to examine the relations among property strategies, working memory, and multiplication tasks with 101 Chinese fourth-grade students. Two multiplication property strategies (associative and distributive) were compared with no strategy and demonstrated differentiated effects on students' accuracy and reaction time. Associative…

The equipartitioning learning trajectory (Confrey, Maloney, Nguyen, Mojica & Myers, 2009) has been hypothesized and the proficiency levels have been validated through much prior work. This study solidifies understanding of the upper level of co-splitting, which has been redefined through further clinical interview work (Corley, Confrey &…

Although learning mathematics certainly depends upon accurate understanding of the facts of multiplication, it requires much more. This study examines the relationship between a meaningful understanding of arithmetic operations and the mastery of basic facts. The study began with a joke about a mistaken mathematical fact. The children appreciated…

Mastering single-digit arithmetic during school years is commonly thought to depend upon an increasing reliance on verbally memorized facts. An alternative model, however, posits that fluency in single-digit arithmetic might also be achieved via the increasing use of efficient calculation procedures. To test between these hypotheses, we used a…

The maturation of multiplicative thinking is key to student progress in middle school as rational number, ratio, and proportion concepts are encountered. But many students arrive from the intermediate grades and falter in developing this essential disposition. Elementary students have historically learned multiplication and division as operation…